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    cauchy sequence

    Show that if  x_n and  y_n are Cauchy sequences then both  x_n+y_n and  x_n y_n are Cauchy sequences
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    Quote Originally Posted by flower3 View Post
    Show that if  x_n and  y_n are Cauchy sequences then both  x_n+y_n and  x_n y_n are Cauchy sequences
    The first is simple noting \left| {(x_m  + y_m ) - (x_n  + y_n )} \right| \leqslant \left| {x_m  - x_n } \right| + \left| {y_m  - y_n } \right|.

    For the second we know that Cauchy sequences are bounded.
    \left| {(x_m y_m ) - (x_n y_n )} \right| \leqslant \left| {(x_m y_m ) - (x_n y_m )} \right| + \left| {(x_n y_m ) - (x_n y_n )} \right|  \leqslant \left| {y_m } \right|\left| {x_m  - x_n } \right| + \left| {x_n } \right|\left| {y_m  - y_n } \right|
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